I'm not going to go over all your simplification but assuming you are correct up to this point, you go astray in your last simplification. The equation becomes

Note that x^2 is in the denominator. The way you had it is NOT a bessel equation. So now that we have this, what do we do?

Well, we know that the solutions to the following equation

are

and

So let's compare this to your equation.

These are our equations

We find the solutions to be

Therefore, the part of the general solution that they want is

Where

Of course K is a constant so we can let a new constant A absorb that value.